Physica 29D (1987) 95-127
North-Holland, Amsterdam
A SIMPLE GLOBAL CHARACTERIZATION FOR NORMAL FORMS OF SINGULAR
VECTOR FIELDS
C. ELPHICK1., E. TIRAPEGUI 2, M.E. BRACHET 3'4, P. COULLET 1'3 and G. IOOSS 5
1Physique Th~orique, Universit~ de Nice, Parc Valrose, 06034 Nice, France. Unit~ Associ~e au CNRS
2Facult~ des Sciences, Unioersitd Libre de Bruxelles, CP 226, 1050 Bruxelles, Belgique and Unioersidad de Chile, Facultad de
Ciencias Fisicas y Matematicas, Departamento de Fisica, Santiago, Chile
30bseroatoire de Nice, 06003 Nice, France
4Groupe de Physique des Solides, ENS, 24 rue Lhomond, 75231 Paris, France
SLaboratoire de Math~matiques, Universit~ de Nice, Pare Valrose, 06034 Nice Cedex, France
Received 10 June 1986
Revised manuscript received 9 June 1987
We derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing
order parameters in degenerate bifurcation problems. Using an appropriate scalar product in the space of homogeneous vector
polynomials, we show that the resonant terms commute with the group generated by the adjoint of the original critical linear
operator. This leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the
normal form. Explicit examples, and results obtained when there are additional symmetries, are also presented.
1. Introduction
The dynamics at the onset of several instabilities in a physical system undergoing a degenerate
bifurcation near an equilibrium point can often be reduced to the temporal evolution of a simple set of
ordinary differential equations. These so-called "amplitude equations" or "normal forms" describe the
behavior of only those normal modes which are mildly unstable or slightly damped in linear theory. All the
other, strongly damped, normal modes are eliminated from the description. This type of reduction, besides
allowing an obvious simplification of the original problem, is also very useful for classification purposes.
An identical set of amplitude equations is obtained for a class of original problems, it will thus display any
dynamical behavior common to this class in a prototypical way. In this context, note that although the
practical interest of the study of multiple bifurcations decreases with the degree of degeneracy, the study
per se of the corresponding normal forms nevertheless presents a great interest because of the possibility of
picking up very rich nonlinear behavior (see Arneodo et al. [1] and references therein). The present work
deals with the derivation and the characterization of such degenerate normal forms.
Of course the problem of computing a normal form is not new; mathematical references can be found
in, e.g., Arnol'd [3] or Guckenheimer and Holmes [11]. There are basically two methods to derive a normal
form. With the first method [15, 12, 7] one first computes a locally invariant and attractive small
dimensional manifold, the so-called center manifoM, on which the dynamics reduces for large times. Then
a nonlinear change of variables is done to put the small dimensional system into normal form. The second
method (which has been used in nonlinear hydrodynamics since a long time, see [17]) has recently been
greatly clarified, see [8] and references therein. With this method, one systematically expands the original
fields in power of the amplitudes of linearly marginal modes, yielding both the normal form and the center
*Current address: Astronomy Department, Columbia University, New York, NY 10027, USA.
0167-2789/87/$03.50 © Elsevier Science Publishers B.V.
(North-Holland Physics Publishing Division)
C, Elphick et aL / Normal forms of singular vector fields
96
manifold. The crucial point in normal form computations is to find a homogeneous polynomial vector field
of degree k in a space complementary to the range of the so-called "homological operator". The
dimension of the vector space strongly increases with k and direct computations soon become impracticable [11]. Very recently Cushman and Sanders [9] used representation theory of the group sl(2,R) which
leads to the normal form in a constructive way.
As we shall see below, our approach is elementary and more direct since no splitting of the linear
operator L o (see (13)) in the critical subspace is needed.
Let us recall that the resonant nonlinear terms of a normal form are those ones that cannot be
eliminated by a nonlinear polynomial change of variable. Technically this means that they are in the kernel
of the adjoint of the homological operator. This is just the Fredholm alternative which, as is well known, is
independent of the nondegenerate scalar product used to define "adjoint" and "orthogonal" (see section
2.1). In this work we define a scalar product in the space of homogeneous polynomials such that the
adjoint of the homological operator is the homological operator associated with the adjoint of the critical
linear part of the original equation. Nonlinear terms of the normal form are thus equivariant under (i.e.
commuting with) the group G generated by the adjoint of the original linear operator, this is our main
result, given in section 2.2. In particular if the linear critical operator L 0 is diagonal then the whole normal
form is found to be equivariant under G. For example in the case of the Hopf bifurcation G is the group
of rotations in two dimensions and therefore the corresponding normal form is invariant under arbitrary
rotations in the complex plane, property which in turn justifies the widely used argument that the
invariance of the original system under time translations implies the invariance of the Hopf-normal form
under rotations in the complex plane.
The general characterization obtained in section 2.1 allow us to interpret the resonant terms (i.e.
equivariant under G) in the normal form as scalar renormalizations of the linear Arnol'd-Jordan
unfolding and to derive a partial differential equation obeyed by these resonant terms. This gives us a very
fast computational method that we apply to some classical examples in section 2.4 and to a more
complicated one in the appendix. The rest of the paper is organized as follows: In section 2.3 we treat the
case where the original problem has an additional symmetry. In section 3.1 we show how to use our
method for the computation of the unfolding of the normal form. Finally, in section 3.2, we give the
unfolding of the classical examples treated in section 2.4. Results for the special case where none of the
critical eigenvalues has zero imaginary part are also given in the appendix.
2. Normal form of the unperturbed vector field
2.1. The homological equation
We study here evolution problems of the form
dZ =.~(Z),
dt
(1)
where Z belongs to the phase space E = R n and .~(0) -- O. We write (1) in a finite dimensional space, with
the understanding that much of the following analysis can be easily adapted for evolution partial
differential systems, such as those occurring in non-linear hydrodynamical problems. In fact, all we need to
assume is that there exists a stationary solution. In what follows we shall avoid, as much as possible, the
C. Elphick et al./ Normal forms of singular vector fields
97
0
o
,o
o
'0'
t7
Fig. 1. A typical distribution of the spectrum of .L#.
use of explicit coordinates such
a s o~//(z 1 . . . . .
Zn) or &aj = 3~/3zj. To wit, let us define
~= Dz~(0 )
(2)
to be the linear operator E -~ E whose matrix dements are given by 3,~/3Zg.
We assume that the set of eigenvalues of £a is composed of two pieces, one on the imaginary axis and
the other with a strictly negative real part (fig. 1). The decomposition of the space E, associated with this
decomposition of the spectrum of .L~
o, is written [14] as
E = Eo • e
(3)
and the restrictions of Za to these invariant subspaces are denoted by L 0 and L_.
The problem of finding a normal form for (1) reads: "Find a polynomial • in (X, Y) taking values in
E, and a polynomial F in X taking values in Eo, as simple as possible, such that we can write
Z=X+
r+qb(X,Y),
X~E0, Y~E_,
(4)
dX
dt = L°X + F( X) + O((IX ( + [YI)P),
(5)
dY
d---~= L_ Y + N( X, Y) + •([Xle),
where
N ( X , Y ) = O ( I Y I ( ( X t + (YI)),
F(X)=O((XI2),
+ ( X , Y ) = O ( ( ( X [ + IY() 2)
(6)
and P is arbitrarily large (but fixed)". The estimate for N in (6) means that the set Y = 0 is invariant under
the dynamics of (5) up to order P. This is a direct consequence of the form for N in (10)4.
Remark O. More precisely our program is to find a set of polynomials Pj which generate the invariant
algebra of the group given by the flow of the linear vector field L*oX (* denotes the usual adjoint
C. Elphick et al./ Normal forms of singular vector fields
98
operation in E0) such that we can write (see Theorem 3)
dim(Eo)
F(X)=
E
%(Pj)Vi,
i=1
where the a~ are rational fractions of the pj and the v~ are fixed vector valued polynomials.
It is worth noting that in all the examples treated here we can in fact write [9]
n (dim ( E o))
F(X) =
~_~
5~(pj)6 i,
i=1
where fii are polynomials in the pj and vi are fixed vector valued polynomials.
We will say that a singular vector field is in normal form if its non-linear part can be written in any of
the two forms for F given above. This will be equivalent (see section 2) to the fact that the Lie derivative of
F (in particular of vi, 6i) along L ~ X vanishes. Let us note that if L 0 is diagonal and F is in normal form
then also the Lie derivative of LoX + F along L ~ X vanishes.
In what follows we often refer to the normal form of (1) while in fact considering the truncated equation
(5)1:
dX
dt = L ° X + F( X),
(7)
which in fact is the object of fundamental interest for further analysis of the dynamics of system (1).
Remark 1. The manifold in E given by the equation
(8)
is the approximation, up to order IXI P, of a center manifold [15, 12, 7] for (1). It is tangent to the subspace
E 0 at 0 and it has the dimension of E 0. It is clear that, if we use (7) instead of (5)1 (neglecting
O((Isl + [YI)P), this manifold is locally invariant since Y remains equal to 0. It is also locally attracting,
due to the negativeness of the real parts of the eigenvalues of L_.
Remark 2. The manifold in E given by the equation
Z = y+dP(O,y),
Y~E_
(9)
is the approximation up to the order I YI P of the stable manifold of 0. It is tangent to E_ at-0, and for any
initial data on it, Z(t) will relax to 0 exponentially when t ~ oe with the truncated equation (5)2 (without
O(IXIP)).
C. Elphick et al./ Normalforms of singular vectorfields
99
Let us start by defining the Taylor expansions of o~, ~, F, N as follows:
=z°z+ E
k>_2
q)(X, Y) =
~q[X(P),Y(q)],
E
p+q>_2
F(X) = E
(10)
Fp[X<P)],
p>_2
N(X, Y) =
~., Npq[X (p), Y(q)],
p+q>_2
q>O
where F?, o~e are p-linear symmetric in its arguments, and C~pq,Npq a r e p-linear symmetric in the X
variable, and q-linear symmetric on Y. Z (k) stands for the repetition of k identical arguments Z,
equivalent notations are used for X (p) and Y(q). o~-k[Z(k) ] is thus a vector valued homogeneous
polynomial of degree k in the components of Z.
In order to obtain equations for rl)pq,Fp, Npq we need to identify the expansions in (X, Y) of
~ - ( X + Y + ~(X, Y)) and of d X / d t + d Y / d t + D x ~ ( X, Y ) . d X / d t + D y e ( X , Y) . d Y / d t where d X / d t
and d Y / d t are replaced by the right-hand side of (5) (the notation D x ~ ( X , Y) stands for the linear
operator acting from E 0 into E, whose matrix is given by (0q~JOXj)).
The identification at the order 1 in X and Y is just a verification:
(11)
.Sf(X+ Y)=LoX+L_Y.
Next, higher orders p + q > 2 lead to identities of the form
. ~ % q [ X (p), g(q)] - D X ~ q [ X (p), r(q)] . Lo X - DyfrPpq[ X (p), Y(q)]. L Y
= {Fp[X(P)]
q=0
"1-Rpq [ X (p), r(q) ],
(12)
Npq[X(P),Y (q) ] q > 0
where Rpq only depends o n (l~p,q,, Fp,, Np,q,; pt -t'-q' <p + q -- 1}. The strategy is to solve (12) step by step,
starting with p + q = 2, and then increasing p + q by 1 at each step.
In what follows we denote by P0 and P_ the two projections on E o and E_ which commute with ~o
[14]. We have
Po + P
= Id,
PoL_ = O,
P _ L o = O,
"~Po = Lo,
cpp_ = L .
(13)
Eq. (12) may be decomposed, yielding for q = 0,
toPo%o[ g (p) ] - DxPo~o[ S(P) ] • tog-~.'Fp[ S(P) ] -}- eoRpo[ X(P) ],
(14)
L_P_~po[ X(P)l - D x P _ % o [ X(P) ] • L o S = P_Rpo[ S(P)].
(15)
100
C. Elphick et al./ Normalforms of singular vectorfields
For q ~ 0, eq. (12) gives
LoPo~pq [X (p), Y(q)] - DxPo% q [X (p), Y(q)]" L o X - D yPo~q[ X (p), Y(q)]" L Y
= PoRpq[X (p), Y(q)],
(16)
L P_~pq[X(P), Y(q)] - DxP_~pq[X(P) , Y(q)]. L o X - DyP_~pq[X(P), Y(q)]L_Y
= Neq[X(P), Y(q)] + P Rpq[X (p), Y(q)],
(17)
where Rpq is known. Since in (17) there is no restriction on Npq, we can choose any P~Ppq, for instance 0,
and then (17) gives directly Npq. Now the eqs. (15) and (16) are explicitly soluble. They lead to
[eL-tp-~PO[(e-L°tX)(P)]] = eL-tP-RP°[(e-L°tx)(P)]'
d r [e - °Lt Por~pq[(eL°iX) (p),
(eL-ty)(q)] ] =
(18)
-e-LotPoRpq [ (eLotX) (p), (eL-ty)(q)],
hence
P-%0[X(P)] = - fo °°dteL-te Rpo[(e-LotX)(P)],
(19)
POdPpq[X(P),Y (q)] = fo°°dte-LotPoRpq[(eL°ts)(P),(eL-tY)(q)],
q>0,
where the integrals are convergent, due to the exponential decay of e L-t as t ~ oo.
Next we need to solve eq. (14) with Fp as "simple as possible". This equation takes the form
[ P0~0 [ X(P)], LoS] = PoRpo[ X(P)] + Fp[ X(P)],
(20)
where the left-hand side is the Poisson-Lie bracket of two vector fields in E 0. Note that (20) is called the
"homological equation" in the standard mathematical literature [3, 10].
Remark 3. Looking at (17) we might think about the possibility of choosing Npq as simple as possible, by
choosing suitably P ~ p q . This equation is also a "homological equation", slightly more complicated than
(20). It is shown in [3, 10] that the linear operator acting on P_ ~pq has eigenvalues given by the following
combinations:
~-)-
Epr~(o) - Eq,~(t-),
r
I
where ~(~) and h(k-) are the eigenvalues of L 0 and L_, respectively, Pr and ql are integers such that
ErPr = P, EIql = q"
If 0 is not an eigenvalue, we can choose Npq = 0. We cannot avoid 0 to be an eigenvalue for q = 1, but if
the following "non-resonant conditions" are realized:
X(; ) 4= Epr h(O)+ Eqlh(i -)
r
I
(21)
C. Elphick et a l . / Normal forms of singular vector fields
101
for any k, r, l such that
2 < Y~.pr+ ~ " q , < P - 1 ,
r
~ q t v ~ 1,
1
l
then we can find
N ( X , Y ) linear in Y, up to order (IXI +
I YI) p.
(22)
Nevertheless, we have to stress that these conditions are impossible to check in the infinite dimensional
case, since L has in general infinitely many eigenvalues.
2.2. Simple characterization of the normal form
In this section we solve (14) (or (20)) with the simplest possible Fp. Let us introduce some useful
notations. We denote by H k the vector space of homogeneous polynomials of degree k in X ~ E o, taking
values in E o. We denote by ~¢t°k the space and scalar homogeneous polynomials of degree k in X ~ E 0.
We can then write
Hk ='~'k ® Eo-
(23)
Note that the eq. (14) is in fact a linear equation of the form
~¢tk)( Po~ko ) = PoRko + Fk
(24)
in H k. Thus the right-hand side of (24) has to belong to the image of ~ ¢ ~ in H k. This gives a condition
for F k which may be chosen in a complementary space of this image. The idea in what follows is to choose
a suitable scalar product in the space Hk, allowing a simple computation of the adjoint operator of ~¢~k).
Since the kernel of the adjoint d eke* is a complementary space of image (~¢¢k)), a good choice is to take
F k belonging to this space.
In what follows we choose a real basis of E o and write X = (x 1. . . . . xu). For two scalar polynomials P
and Q, let us define
~PIQ> = P(O)Q( X ) l x - o .
(26)
where aj means a/Ox i. For instance we have
t
( x ° l x l') =
( xP ) =
where 8~t~ is the Kronecker symbol. Now (PIQ) defines a scalar product on the space of scalar
polynomials in X. We write it symbolically
(PI Q) = P(O)Q( X) lx_ 0.
(26)
This scalar product was introduced in quantum mechanics and studied in particular by Bargmann [4],
C. Elphicket al./ Normalforms of singularvectorfields
102
explicited by using the standard L 2 scalar product. This basic property is that
(QRIP) = Q(~)R(O)P( X)Ix=o = R(~)Q(~)P( X)Ix=o = (R[Q(O)P),
(27)
i.e. multiplication by the polynomial Q is the adjoint of the differentiation by Q(O).
The corresponding scalar product in ~ k is obtained by taking homogeneous polynomials P and Q of
degree k. We can now endow the space H k of vector polynomials with the natural scalar product
N
(vlw),,k= E ( lwj),
(28)
j=t
where V = (V t .... , VN) and W = (W 1. . . . . WN) ~ H k and Vj, Wj are homogeneous polynomials of degree k
in X = (x t .... , x u ) .
Let us consider a linear invertible operator A in Eo; we then have for any polynomials P and Q,
( P( AX)IQ( X)) = ( P( X)IQ( A*X)).
(29)
In fact, if we set Y = A'X, it is easy to see that 0 x = AOy, hence the right-hand side of (29) is just
P( A~y)Q(Y)[y=o= ( PoAIQ).
Now, by construction of the scalar product in Hk, we have for any V and W in Ilk,
(A -1V( AX)I W(X)) "k =- ( V ( A X ) IA* -XW(X)) ~¢k= ( V(X) IA* -tW(A*X)) Hk,
(30)
where we used (29) to get the last identity.
Let us choose A = e L°t in (30), then we have
(e-L°tV(eL°tg)[ W( g))Hk = ( V( X)le-Lf~tW(eL~tX) )l_lk,
(31)
where we have used the elementary result
(eLo,)* = eL~ t.
Now differentiating (31) with respect to t, at t = 0, we readily obtain
( .~'(k)V I W ) Hk= ( VI~¢*(k)w) H~,
(32)
where
[.~¢,(k)W] [ X(k)] = L ~ W [ X (k) ] - D~.W[X(k)] • L~X
= [W[X(k)],LZX].
(33)
This identity shows that the adjoint of the homological operator s¢ (k) in Hk, built with L0, is the
C. Elphick et al./ Normal forms of singular oectorfields
103
homological operator in Hk built with the adjoint L6~. So we have proved the following:
Theorem 1. The following decomposition holds for the space H k of homogeneous vector polynomials of
degree k in X ~ E0:
H k = Im ~¢(k) ~ Ker ~¢,(k),
where
Kerd(.k)=(VEHk;e-L~tV[(eL~tx)(k)]
= V[x(k)],
Vt~R,
VX~Eo}.
Summing up this result for all p, we obtain:
Theorem 2. Generalcharacterization of the normalform. A normal form F( X) of the nonlinear terms of (7)
can be found such that F commutes with exp(L~t), t ~ R, where L~' is the adjoint of L 0 in the invariant
subspace E o. An equivalent characterization is the partial differential system
DxF (X). L~X- LZF(X) = 0.
(34)
Remark. We note, after (31), (32), that
[exp (~¢,(k't)" ~/'][ x(k)] :eL~tV[(e-LJtx)(k) l •
(35)
It is an elementary result that
( exp (~¢,~k)t) • V independent of t } - ( V ~ Ker ~¢,~k) }.
Corollary 2. If L o is diagonalizable, then
Ker(~¢,(k) ) = Ker( d (k))
and a normal form may be found such that it commutes with exp(Lot), t ~ R.
Proof. If L o is diagonal in a suitable basis (complexified Eo) , then L~' = - L o . Hence the result is obvious.
Example. Let us assume that L o has two pairs of simple pure imaginary eigenvalues +i~oo, +io~1, and let
us define X = (z0, ~0, Z1,Z1)" The components of the normal form are sums of monomials of the form
z~qz(~. It is easy to see that
e L J t s = (e-i~°otz0, ei~°ot,~o, e-i~oltzl, ei~°lt,~l) '
hence the commutation property of F,
e-L8tF(eL8tX)=F(X), Vt~R, VX~Eo,
(36)
C. Elphick et al./ Normal forms of singular vectorfield
104
leads, for its first component, to coefficients such that
Wo(p-q-1)+w,(r-s)=O
(37)
and for the third component,
w,(p-q)+w,(r-s-1)=0.
(38)
We immediately see that if o,,/wi is irrational, we only need to keep in the first component the coefficients
such that p = q + 1, r = s and in the third one r = s + 1, p = q. This shows that in this case the normal
form is equivariant under the group T*, and the trajectories {eLz’X, t E W} are dense on the torus T*.
Now, if oO/wl = m/n, where (m, n) = 1 (m and n have no common divisor), then the coefficients of the
first component of the normal form are such that
p=q+l+kn,
r=s-km,
kEZ.
(39)
This shows that the normal form (7) is (2 complex equations)
dz
(40)
where P,, P,, Q,,, Q, are polynomials in their arguments.
Hence the normal form is in this case invariant under the subgroup of T*: (da, 0,) + (0, + ms, 8, + ns),
s E IR (- T’).
Remark.
In this example, corollary 2 applies.
Another very fruitful way to compute the normal form F(X) is to use the linear partial differential
equation (34) satisfied by F. Choosing a basis in E,, complex in general when we use the Jordan
decomposition of L,, this equation may be written more explicitely as follows:
(41)
The characteristic
dx/
czo,,x,
system [20] associated with (41) is the following:
dq
= cEo,iF,,
I
i,_i=I,...,N
I
and the characteristic
curves are the trajectories in E, defined by
{eLgfX; tER}.
So that, if we write (42) = dt, we immediately recover (36).
(42)
C. Elphick et aL/ Normal forms of singular vectorfields
105
Now, instead of computing exp (L~t) and using (36), we can use the independent first integrals of (42).
For this we need to put L~' into Jordan form (this also facilitates the computation of exp (L~t)). Then to
compute the normal form we can use the following result:
Theorem 3. Representation of the normal form. The normal form F(X) of theorem 2 may be made explicit
as follows:
N
F( X) = Y'~ aj( X)~jX,
(43)
j=l
where .~j, j = 1 . . . . . N are linear operators commuting with L~' in E0, defined by (44), and the scalar
functions %, j = 1,..., N are rational fractions, first integrals of the characteristic system d X/dt = L~X.
Remark. It is easy to give a little more precise characterization of the rational fractions %. See the proof
of the theorem above and Appendix A.2 for the explicit form of the %.
Proof of theorem 3. Let us choose N linear operators .o¢41. . . . . .oqaN commuting with L~ (they then belong
to Ker ~ ' .(1)) such that for almost all X, the system (LPjX; j = 1,..., N} forms a basis of E o. Assume that
L 8 is in Jordan form, with r blocks, each block L~'j corresponds to an eigenvalue hj (Re)~ = 0) and to an
invariant subspace E0j. Define the projection Pj on Eoj such that
=Ideo
j=l
and denote by pj the dimension of E0j. Then, the linear operators
Pj,(L;j-~j)Pj
,(L~j-Xj)p'-IPj,
....
j = l .... ,r
(44)
are N linearly independent linear operators .Wk commuting with L~'. It is easy to check, that if X has a
nonzero first component in each Eoj, then the system ( ~ j X ; J -- 1. . . . . N} space E 0. We can then write
(43) for almost all X in E 0.
Now, since
j=l
. . . . . N,
we also have
eLJt.~j = ~ j e L~t,
j = 1 . . . . . N,
hence the commutation relation of F (36) with e LSt leads to
etj(eL'~tX)=otj(X),
j = l . . . . . N.
So, the functions aj are first integrals of the characteristic system. Now, F ( X ) is a vector polynomial in X.
Computing the decomposition (43) of F ( X ) on the basis (.~jX; j - - 1. . . . . N } we easily observe that the
106
C. Elphick et al./ Normal forms of singular vectorfields
coefficients are rational fractions of the components of X whose denominators are powers < ~,j of the first
component of each E0j space (see Appendix A.2).
Remark A*. The simple examples treated in section 2.4, all lead to the following type of normal form:
F( X) = ~_,flj( X)Vj + Y'~aj( X ) ~ j X ,
J
(45)
J
where the scalar functions a d and flj are polynomial first integrals of the characteristic system, and where
(Vj} are eigenvectors of L~' while ( ~ } are all the linear operators commuting with L~' (Ker~¢,~l)). In
fact, theorem 3 shows that we can always find the aj's as rational fractions. For instance, in the normal
form (40) the possible denominators are z 0 and z v Note that if we want to express in this example the aj's
as polynomials of the scalar invariants Zo$o, z1~1, z~$~', E~z~' then the possible denominators are z0E0 and
zl$1. We shall see a more complicated example in Appendix A.2.
When (45) holds with polynomials aj and/3:., it is possible to give an obvious physical interpretation:
the effect of the nonlinearities is simply to renormalize the increasing rates, and linear coupling terms (see
Arnol'd [3] for the affine normal form). This renormalization is done via scalars of the group exp (L~t). It
is nice that this holds for all low codimension examples given in section 2.4.
2.3. Case of an additional symmetry
Let us assume that the system (1) is invariant under the representation of some symmetry group. The
simplest case is when we only have a symmetry S ~ Id such that S 2 = Id.
For the moment let us assume that we have a linear invertible operator T in E such that
~(TZ)
= T~(Z).
(46)
Then T commutes with Za, Lo, L_, and with any derivative of ~- at the origin. Hence it is clear that To
and ~¢(*) commutes in the following sense: let us define for any V in H k,
To,V[ X(k)] = T o l V [ ( T o X ) ( k ) ] ,
To= T[E O,
then it is easy to verify that
It follows that the image of ~¢<k) is invariant under To,, as well as the kernel of ~¢<k).
Let us moreover assume that
TO= TIEo is a unitary operator on E 0,
(47)
i.e. To* = TO 1 on E 0, then TOcomputes with L~' since
TolLo = ZoTo I leads to
L~T o = ToL ~.
As a consequence, on H k the linear operator To, commutes with .ae,(k), hence the kernel of ~¢,(k) as well
c. Elphick et al./ Normal forms of singular vectorfields
107
as the image of ~¢(k) are invariant under To,, which commutes with the projection operators defined in
theorem 1.
It results that the normal form defined in theorem 2 commutes also with T0. We have proved the
following theorem:
Theorem 4. If there exists a linear invertible operator T which commutes with the vector field ~ , and if
its representation TO on the invariant subspace E 0 belonging to the eigenvalues of zero real part is a
unitary operator, then a normal form can be found which commutes with TO as well as with e L~t, t ~ R.
If all eigenvalues of L 0 are semi-simple ( L 0 diagonalizable), the additional assumption on TO is useless,
because we can avoid to project orthogonally onto Ker.a¢.(k), by projecting in a more natural way on
Ker .a¢(~) which, in this case, is complementary to Image (~¢(k)) (see [13]). The natural projection is then
defined for any V in H k by
P(V)= lim --1 fo (e L°t ) , V d t .
(48)
,r...~ o O "r
We have just showed the following:
Corollary 4. In the case when L 0 is diagonalizable, and if there exists a linear invertible operator T
commuting with the vector field ~ , then a normal form may be found which commutes with TO as well as
with the group e Lot, t ~ R.
For an example with a symmetry and a non semi-simple eigenvalue see section 2.4.3.
2.4. E x a m p l e s
Note that examples 2.4.1, 2.4.2, 2.4.3, 2.4.4 are also treated by Cushman and Sanders in [9], using a
different, very elegant method. Our method is seen to lead to more elementary and shorter computations.
2.4.1. ~-2 singularity (see [2, 8])
We have
Zo=
(o
0
hence X = (x, y ) and F ( X ) = (Fl(x, y), FE(X , y)). Eq. (34) may be written as
-if=0,
This leads to the solution
F1Cx, Y) = Xf~l(X),
F2(x, Y ) =YrPlCx) + CPE(X),
where I]01 and cp2 are polynomials (see Appendix A.3 for the proof).
(49)
108
C. Elphick et aL/ Normal forms of singular vectorfields
This normal form is in the frame of remark A* of section 2.2, (see (45)) with
and
~x(X) = ~l(x),
~2(x) = B(x) + ~2(x)
For terms of degree k, we obtain a subspace of H k which is 2-dimensional:
( ax k, ayx k-1 + bx k)
(50)
Note that we can change the normal form by choosing another projection (no longer orthogonal in Hk). In
fact if we add to (50) the term ( - a x k, k a x k - l y ) , which is orthogonal to Ker~¢,~k) (hence it is in
Image (~¢,~k))), this leads to a simpler normal form:
( O , a ' y x k - l + bxk) •
(51)
Hence it is possible to write (7) for this example in the form
dx
a t --Y'
dy = xyPl(x ) + x2P2(x),
d-7
(52)
where P1 and P2 are polynomials in x.
2.4.2. ~3 singularity
This example is also treated in [9]. Here we have
Lo=
0
0
0
0
•
Let us denote by x 1, x 2, x s the three components of X and by F1, F 2, F 3 the 3 components of F. Then
(34) is written as follows:
a~l
aF1
xl a-~ + x 2 ~ s = 0,
OFz
OF2
x1~-~2 + x 2 - ~ -3 = F 1,
OF3
(53)
OF2
X l '~--~2 -4- X2~--~3 = F 2 .
Here, the characteristic system is
dXl
dx2
dx3
dF1
dF2
dF3
-'0--= x----~"= x z =--0--=-'~-1 = F 2
(54)
C Elphick et al./Normalforms of singular vectorfields
109
and the first integrals are
x,,
F a, x x r 2 - x z r 1,
x22 - 2xax 3,
x l r 3 + x 3 r 1 - x z F 2.
(55)
Hence, the general solution of (53) takes the form
r l ( X l , x2, x3) = Xl~ 1(X1, x 2 - 2XlX 3),
F2(x1, x2, x3) -~ x2~ 1( x1, x22 - 2XlX 3 ) -t- Xl~2(Xl, x 2 -
2xlx 3),
F3(x1, x 2 , x 3) = X3q01(X1,
2XlX ,) + % ( x l , X22- 2 x l x 3).
X 2 --
2XlX 3) + x2ep2(X1,
X 2 --
(56)
It is not very hard to show that qoj, j = 1, 2, 3 are polynomials in their arguments (Appendix A.3), so we
remark that (56) enters into the framework of remark A* of section 2.2, provided that we write
0
t]03
=x(i
-t- 013( X ) & X,
~3 =
0
,
0
where a 3 and fl are polynomials in X, ands La3L~' = L~'LP3.
Here, like in example 2.4.1, we can change the projection, and choose a normal form such that (7)
becomes
dxl
dt = x 2 '
dx 2
dt = x 3 '
(57)
dx 3
-dt = x 3 P l ( x l ,
x 2- 2XlX3) + x2P2(xI,
x 2-
2x1x3) + P 3 ( x I , x 2 - 2x,x3),
where Pj, j = 1, 2, are polynomials in their arguments starting at degree 1, while /'3 is a polynomial
starting at degree 2.
2.4.3. f2~2 singularity
This example is also treated in [9]. Their result, once simplified as in (62), is the same as ours. Here we
have
0
1
00'
0
0
0
0
0
0
0
1
0
0
0
0
Lo=
Let us denote by xj and Fj, j = 1, 2, 3, 4 the components of X and of F in the normal form (7). Then the
characteristic system (20) becomes
dXl
0
d x 2 = dx3 = dx__A- dF1
x1
0
x3
0
dF2 -
el
dF3 - dF4
o
f3
(58)
110
C. Elphick et a l . / Normal forms of singular vector fields
First integrals are given by
X 1,
X 3,
F 1,
U=X2X3--XIX4,
F3,
xIF 2 - x2F1,
(59)
x3F 4 - x4F3 ,
Hence, the general solution of (41) is here
F~(x~,~,x,,~.) =x~dx~,x,,o) +x,~dx~,x,,o),
F2(xl, x2,x3, x4) =X2+l(X1, X3, V) +x4+2(Xl,X3,u ) ++3(Xl, X3),
F3(Xl, X2,X3,X4) =x3~4(x1, x3,0) +Xl~5(x1,x3,0),
F4(x1, x2, x3, x4) =x4~4(Xl, X3,0 ) +x2~5(x1,x3,u ) +~6(Xl,X3).
(60)
Since {Ffi j = 1,2,3,4} are polynomials in Xl, x2, x3, x4, it is not difficult to show that cpj; j = 1,... ,6 are
polynomials in their arguments, i.e. in x 1, x 3, v or in xl, x 3. This means that this normal form enters into
the frame of remark A* of section 2.2, since
X1
=
~l(X)
+~3(x)
X2
0
1
F(X)
0
X1
"}- 0/1(X)
q- ~ 2 ( X )
-{- o~2(X )
0
0
0
0
0
1
0
0
0
0
X3
0
+ ~.(x) o + ,~(x)
X4
X3
X4
X1
+~7(x)
"Jr 0~6( X )
0
0
0
0
0
0
0
Xl
X3
0
0
X2
+as(x)
0
(61)
0
X1
where
O~l(X)---rpl,
~5(X)=q02,
o~3(X)=qo 4,
o~7(X) =qo5,
~ l ( X ) + XlO/2(X ) + x3R6(X ) = q93, /~2(X) + xlo~8(X ) + x30t4(X ) = qo6.
Now, as in examples 2.4.1 and 2.4.2, we can change the projection, and choose the following normal
form for (7):
dx 1
dt -- x2'
dx2 = x z P 2 ( x l , x3, x2x3 - XlX4) -4- x 4 P z ( x l , x3, x2x3 - x l x 4 ) + Q l ( x l , x3),
dt
(62)
dx3
dt = x4,
dx 4
dt
where P1,
gree 2.
- x 4 P 3 ( x l ' x3, XzX3 - x l x 4 ) + x 2 P 4 ( X l ' x3, x z x 3 - x l x 4 ) + Q z ( X l ' x3)'
P2, P3, P4, Q1, Q2 are polynomials'in their arguments, Pj starting at degree 1, and Qj at de-
C. EIphick et aL/ Normal forms of singular vectorfields
111
Let us assume now that the vector field ~ commutes with a symmetry S. Let us also assume that S is
not trivial on E o in the following sense: we can choose the eigenvectors of L o such that they are exchanged
by S, as well as the two generalized eigenvectors. The matrix of S in the same basis as for L o, is now
0
0
1
0'
0
0
0
1
1
0
0
0
0
1
0
0
S=
S is clearly a unitary operator on Eo, hence the results of section 2.3 apply. We can then find a normal
form invariant under S by taking
P3(Xx, X 3, x 2 x 3 - x l x 4 )
= P I ( X 3 , x 1, XaX 4 - X 2 x 3 ) ,
P 4 ( x l , x3, x 2 x 3 - XlX4) = e2(x3, Xl, x 1 x 4 - x2x3) ,
(63)
a 2 ( x l , x3) = Ox(x3, x l ) .
2.4.4. ~o2 singularity
This example is also treated in [9], where the normal form is written differently. The physical motivation
of such a singularity is discussed in section 3.3.4. Here we have ( ~ ~ 0)
i¢0
1
0
0
0
ito
0
0
0
0
-i~
1
0
0
0
-i~0
Z0=
Let us write X = (zl, z 2, zl, z2) and F = (F1, F2, F1, Fz) , then the characteristic system (42) becomes
dZl =
dz2
-i~°z,
zl - i°~z2
=
d;T1
1~°zl
_
dz2
_ dF1
~1 + 1°~z2 -i~°F1
dF 2
F1 - iwF2"
(64)
Now, first integrals are
Z1Z1,
ZIZ2 -- ZIZ2 ,
io~z2/z I + In Zl,
ziF1,
ziF2 - z2F1.
(65)
The general solution for F 1 may then be written as follows:
Fl(Zl, z2, Zl, z2) -~-ZlfPl(ZIZ1, ZlZ2 - ZlZ2,i°)z2//Zl q- In zl).
(66)
Using the fact that F 1 is a polynomial, it is not very difficult to deduce that opt is a polynomial of
zaS~, Zl~ 2 - ~1z2, moreover independent of its last argument. By an easy argument it can be shown that F 2
takes the form predicted by remark A* of section 2.2:
F2(Z1, Z2, Z1, Z2) =Z2~I(Z1Z1, ZIZ2--ZIZ2) +ZI~2(Z1Z1, Z1Z2--Z1Z2) ,
(67)
C. Elphicket al./ Normalforms of singularvectorfields
112
where cp2 is again a polynomial in its arguments. Finally, adapting the projection on a suitable space as we
did in previous examples, we obtain
dz 1
dt = i~°zl + z2,
(68)
dz 2
dt = i, z2 + zl 01(zl l, zle2 - e z2) +
zle: -
with cpa and ~t)2 polynomials in their 2 arguments (see Appendix A.3). On this system it is tempting to
make the following change of variables:
yj=e
--i~t
zj,
j=1,2.
Then we get in C 2,
dYt
- -dt
= Y2,
(69)
dY2
dt = YlCPl(Ytfil, YtY2-
YlY2) + Y2tP2(Y,.~I, YtY2- Y,Y2),
which is a simpler autonomous second order differential equation.
3. N o r m a l f o r m of a vector field perturbed near a singularity
3.1. General computation
We consider now a system depending on a parameter # = (/~1. . . . . /~,,) ~ R'% of the form
dZ
dt = ~ - ( b t ' Z)
(70)
where ~ is supposed to be regular with respect to (/~, Z) in a neighbourhood of 0 in @" × E. We assume,
as before, that
~(0,0)
= 0
(71)
and we write as in section 2.2
(0, 0)
where Sg satisfies the same properties as in section 2. We want to obtain a normal form for (70). The idea is
to find polynomials ~(~, X, Y), N(g, X, Y), F(Iz, X) such that F is as simple as possible and where we
can write
Z=X+
Y+dP(bt, X , Y ) ,
X~Eo,Y~E_;
(72)
dX
dt = L ° X + F(Ix' X ) + O[([IL[ + IXI + IY[lJ']'
dY
d--i- = L Y + N(t~, X, Y) +
q~(F, X, Y) =
O(l~tl +
(73)
0[(1~1 + ISl)~'] ;
(IXl + I YI)2),
N(/~, X, Y) = O[I Yl(Ittl + ISl + I YI)]~
F(t~,O) = O(I/~l),
DxF(0,0) = 0 .
(74)
C. EIphicket al./ Normalforms of singularvectorfields
113
The estimate for ~/i in (74)1 is due to the absence of linear terms in • and to the fact that
• (#,0,0) = O([#l ). The estimate for N in (74)2 means that the set Y = 0 is invariant under the dynamics
of (73) up to order O[(l# I + [XI) p] and is an immediate consequence of (80) 4. In what follows we show
that F(~t, X) can be chosen such that it commutes with exp (L~'t), just as in the case where there is no/~.
In particular,
F(/z,0) ~ KerL~'
(75)
DxF (/~, 0) commutes with L~.
(76)
and
In the following we shall often refer to the "normal form"
dX
dt = L ° X + F(#, X).
(77)
Remark 1. The manifold in E given by the equation
Z = X + ~(/~, X,0),
X~E 0
(78)
is the approximation at order ([/~l + [ s [ ) p of a center manifold for (70) (see remark 1 in section 2.1).
Remark 2. If F(/~,0)= 0, the manifold in E given by
Z = Y + q~(/~,0, Y),
Y ~ E_
(79)
is the approximation at order (1/~1 + i Yl)e of the stable manifoM of the fixed point X = Y = 0 for the
system (73) without the terms of order P. If F(#, 0) ~ 0, this is not a fixed point and the manifold (79) has
no special meaning for (70).
Let us now start by defining the Taylor expansions of all the functions introduced above, as in section
2.1 (same notations):
~ ( ~ ' Z) =
E ~pq[~(P), z(q)],
p+q>l
~b(/X' X, Y) = qblOO[bt] +
E
F(I-£, g ) =
p+q>l
~"01= ~ ,
E
~qr[lX(P),X(q),y(r)],
p+q+r>_2
.Cq[~(P), g(q)],
(80)
(p, q)~(0,1)
N(/I, X, Y) =
E
p+q>_l
r>_l
Npqr[~ (p), x(q), z(r)].
We now identify terms of the same degree in (/~, X, Y) in (70) where we replace Z by (72) and d X / d t ,
d Y / d t by (73). For r = 0 we obtain
,~pqO[~(P),X (q)] -Dx~pqO[I,L(P),X(q)] .LoX=Cq[I~<P),X<q) ] + Rpqo[~(P),X(q)],
(81)
where Rpq 0 is a known function of @p,-1,q,+l,o, @p,,q,,O, Fp,q, with p' _<p, q' _< q and p' + q' _<p + q - 1.
114
C. Elphick et al./ Normal forms of singular vectorfields
F o r r > 1, we obtain
~pqr [IX(p), x(q), y(r) ] _ D xdppqr [p(p), x(q) ' y(r)1, t o X _
D y~pqr [/x(p) ' x(q) ' y(r) ] t - r
= Npqr[~(p), g(q), y(r)] + Rpqr[~(p), x(q) ' y(r)],
Rpq r is a
<_p+q+r-1.
where
k n o w n function of
(82)
~p'-l,q,+X,r', ~p'q'r', Fp,q,, Np,q, r,
with p ' _<p, q' _< q, r ' _< r, p ' + q' + r '
T h e strategy is first to c o m p u t e the coefficients for p = 0, and then to increase the values of q + r,
starting with q + r = 2, as in section 2.1.
W e then c o m p u t e for p = 1, starting with q + r -- 0, and so forth c o m p u t i n g for a fixed value of p all the
needed coefficients by increasing the values of q + r, starting at 0.
W e note that p = 0 gives the same computations as the ones of section 2.1. This determines Foq, ~Oq, Noq~
for any (q, r). Now, when p ¢ 0, we remark that eqs. (81) and (82) have the same structure as (12), hence
Fpq will have the same structure as Foq, and the determination of ~pq, p >_1, is the same as for ~Oqr, the
arbitrariness on Npq r being the same as on Noqr.
Let us consider the special cases when r = 0 and q = 0 or q = 1. If q = r = 0, then (81) reduces to
•.~(/)p00 [~ `p)] =
Fpo[l~(P)l + RpOO[I~(P)l .
W e saw in section 2.2 that
Fpo m a y
(83)
be chosen in Ker ~¢,(o) and since W ~ H 0 is independent of X,
Ker ~,,(0) = { W ~ E 0; L~' W = 0 } = Ker L~' = Ker £~0,
Then, we recover a well-known result of linear algebra, i.e.
Fpo E Ker L~' ~
eL~Fpo =
Fpo.
If q = 1, r = 0, then (81) reduces, once projected on E0, to
goPot~plO[~ (p), X] - eofI)plO[t£`p), L o X ] = Fpl[lt (p), X] nt- PoRplo[]l, (p), X].
W e solve this equation by choosing
Ker ~ ' , (~) =
Fpl in
Ker ~¢,(1), and since W e H t is a linear operator in Eo:
(W~L#(Eo), L ~ W - WL~ =
0},
i.e. Fpl c o m m u t e s with the operator L~' (hence with exp
this case for V, W ~ H 1 we have in fact
(VI
W)H~ =
Tr (VW*) = E ~jW~j
and it is clear that
(LoV- VLol W ) H 1 =
(VIL~W-
(84)
WL~).
L~t).
We recover a k n o w n result (Arnol'd [3]). In
C. Elphick et al./ Normal forms of singular vector fields
115
We have then proved the following:
Theorem 5. Normalform of theperturbed vectorfieM. A normal form F(/~, X) in (77) can be found such
that (72)-(74) are satisfied and
F(l~,eL~tX)=eLJtF(tt, X),
X~Eo, t ~ R ,
(85)
where L~' is the adjoint of L 0 in E 0. In particular F(/~,0) is in the kernel of L~' and D x F ( / t , 0 ) commutes
with L~'.
Remark 3. If L 0 is diagonalizable then we can find a normal form F(#, X) which commutes with
exp (Lot). (See corollary 2.)
Remark 4. In the case of an additional symmetry, the result of theorem 4 still holds for F(/~, X).
Remark 5. The form (73) of the system is very useful to study in a simpler way the dynamics of (70).
Nevertheless the terms ~[(1~1 + IXI + I YI) e] may give rise to serious problems even for very large P. It is
fortunate that these terms can be simplified in the case where 0 is not an eigenvalue of £#. We already
know that F(/~, 0) = 0 by construction, but in fact we can prove the following:
Theorem 6. If 0 is not an eigenvalue of £#, then we can find a normal form of the perturbed vector field, of
the same type as (72)-(74), but where ~[(1#1 + ISl + I YI) P] in dX/dt is replaced by 0[(IXI + I Yl)e],
and F(/t,0) = 0.
The proof of this theorem is given in Appendix A.1.
3.2.
Examples
Hereafter we consider the same examples as in section 2.4, with an additional parameter # =
(~1 . . . . . ~ m ) E R m
3.2.1. ~2 singularity
The notations are the ones of section 2.4.1. The kernel of L~' is one-dimensional so we can redefine/~1
by setting
The linear operators which commute with L~' are
(: :)and (::
116
C. Elphick et aL / Norrnal forms of singular vector fielck
hence we can redefine the parameters in such a way that
F(/*, X ) = (
/*2x
+/*2Y
+ h.o.t, in X.
(86)
\ /.1 +/.3 x
Remark 6. Here /.1, ~2,/.3 are in fact functions of the original /* ~ R m not necessarily its three first
components. These functions are at the leading order linear combinations of the components of/*. Finally
changing the projection as in section 2.4.1, we obtain the normal form (77)
dx
dt
dy
dt
=y,
(87)
=
/*1 "}- 1'£3x "['-/*2Y + xYP1(/*, x) +
x2P2Ckt,x),
where /*=(/.1,/.2 . . . . ) and P1 and P2 are polynomials in their arguments. Now, making a small
translation in x, we can generically suppress the term /~3x: it is sufficient for that to have a non-small
coefficient of x 2 in (87). We then obtain the classical normal form [18, 2, 3]. Since there are two
fundamental parameters /.1 and /~2 here, one says that this is a codimension 2 singularity. Other
components of/~ play a minor role, changing slightly the non linear coefficients.
3.2.2. ~3 singularity (see [3] for the linear terms in X )
In the same way as above, it can be easily shown that the normal form is obtained by adding to (57)3 the
affine terms (see Remark 6 for the redefinition of the/.i)"
/*0 + / * l x l +/.2x2 +/.3x3~
(88)
and by considering that the coefficients of polynomials Pj, j = 1, 2, 3 are functions of/*. Moreover, as in
3.2.1, we can generically suppress/*aX~ by making a small translation on x v In the same order of idea as in
3.1.1, we shall say that the ~3 singularity has a codimension 3 (main parameters: /*0,/.2,/*3).
3.2.3. ~2ff2 singularity with an additional symmetry S
The kernel of L~' is two-dimensional, but due to the invariance under S we have here (redefining/*o)
0
F(/*,0) =/*0
1
0
1
The kernel of ~¢.m is 8-dimensional, but thanks to the invariance under S we will have to add in (62)2 and
(62) 4 respectively (see [3] for the case of a non-symmetric linear part):
/*0 q- ~ l X l "[- /.2X2 "[- /.3X3 q'- /.4X4,
(89)
/*0 q- /*3Xl + /.4X2 q- /.1X3 -{- /.2X4 •
In fact we can here again generically suppress t~1 and /*3, by making a small translation in x I and x3,
which commutes with S. Hence we have here a codimension 3 singularity, since the role of the other
C. Elphick et al./ Normal forms of singular vector fields
117
components of I* just slightly modify the coefficients of polynomials P1, P2 and Q1. Note that without
symmetry this singularity is of codimension 8.
3.2.4. w2 singularity see [5, 9, 10, 16]
We use the notations of section 2.4.4. The kernel of aacm.is 2-dimensional with complex coefficients,
hence 4-dimensional with real coefficients. We have to add to (68)2 the term
(90)
~1Z1 + ]3,2Z2
on the right-hand side, where/~1 and/*2 are complex. Other components o f / t occur in higher order terms,
at least cubic, in the polynomials cpl and cp2.
In fact, the relevant parameters are Re/t 2 and /*2+4/* I (complex), hence we shall say that this
singularity is of codimension 3: two parameters are necessary for having two pairs of eigenvalues crossing
the imaginary axis simultaneously, and one parameter to have them crossing at the same point.
A special case is when we have a conservative system. This is well known in mechanical problems like the
forced oscillations of a wing under the aerodynamical effect of the wind. Here Re/~ 2 = 0 since volumes are
conserved. Now, the interesting situation (Hamiltonian system) is when two pairs of pure imaginary
eigenvalues, moving as a function of a real parameter, meet together at /~ = 0. Then, they escape
orthogonally from the imaginary axis. A unique real parameter/~ is sufficient to describe such a singularity,
which in this context is only of codimension one.
Acknowledgements
The authors wish to thank A. Cerezo, P. Chossat and F. Rouviere (Lab. of Mathematics of Nice
University) for their constructive remarks. We also thank V. Arnol'd and one of the referees for bringing
refs. [19] and [21], respectively, to our attention.
Appendix
A.1. Case when 0 is not an eigenvalue
Here we want to prove theorem 6. So, we assume that 0 is not an eigenvalue of L 0. A first consequence
is the existence of a persisting fixed point Z0(/, ) regular function of/*. To compute it, it is sufficient to
identify the powers of it in
(91)
where Z is replaced by
(92)
Zo( ) = E
p>_l
and ~
by (80)1. With the notations of (72) we have
(93)
z0(t,) =
Let us now set Z = Z0(/, ) + 2~, then
d2
dt = ~ ( / * ' 2~) = # - ( p , Z0(/x ) + 2~),
~ ( / * , 0 ) = 0.
(94)
118
C. Elphick et aL/ Normal forms of singular vectorfield~
Let us denote by
~g = D2~,~(g,O) = D z , ~ ( g , Z o ( g ) ) .
(95)
the derivative of ~ at the fixed point. We want first to show how to decouple X ~ E o and Y ~ E in the
linear terms in 2~. We remark that we want more than (73) since we do not want any more 0(1#1~'-XlX[)
terms in the right-hand side of d X / d t and d Y / d t (we already have no terms of O(Igle)).
In fact we want to find linear operators ~lo(g) ~ S ° ( E o , E ) and ~bol(g ) ~&a(E_, E ) such that for any
X ~ E o and Y ~ E _ we have
.£#~,(X + Y + Cb,o(g)X + Ckoi(g)Y) = L(~°)X+ L~-)Y + ,~o(g)L(o)X + ¢o~(g)L(-)y,
(96)
where
L~°' = Lo + o(1~,1) ~ z e ( e o )
and
t~->=t +O(I~I)~.~(E).
After projecting (96) on E o and E_, we obtain
Po*ox t ( - ) -
Pooo-°9/~eo*ox= P o l i t e _ + eo,.~P ~J01,
(97)
L~-) + P-¢olL~ ) - P-o~.P-¢ol = e _ ~ . e _ + ?_Ze~eo*ox
and
P-•lO t ( 0 ) - P - , ~ g P - ~ l O = P-,~qggeo + e-o~p, eo{/)lo,
(98)
/.~o~ + ? o , loL~O~_ ?oZO ?o01o = eoZe.eo + ?oZe.e_ 01o.
Since there is no restriction on L~-) ~ ( E _ ) ,
we can choose P - ~ O l = 0, hence the system (97) reduces
to
eo~Jol(P_~t~P_ + P_.~#eo~ol) - Po~gPo~ol = Po,~gP-
in £ a ( E _ , Eo) ,
(99)
where the only unknown is Po~ol(/~) ~ S a ( E _ , Eo).
We now observe that
P L#~P_=L_+0(Igl),
P-,L#~,Po = o(1~1),
Po£#,Po=to+O(Igl),
eo-W~P- = o(It~l),
hence (99) takes the form
g(Po~o,, g) = 0 in .W(E_, Eo) ,
(100)
C. Elphick et al./ Normal forms of singular vector fields
119
where
g(0,0)=0,
Dlg(0,0).A=AL_-LoA
inoLP(E ,Eo)
for any A ~.W(E_, Eo). We already saw that the linear operator Dig(0,0) which acts in ..~e(E_, Eo) is
invertible, since we computed explicitely its inverse in (19)2 with p = 0, q = 1. Hence the implicit function
theorem applies to solve (99), which means that we can compute, by identifying powers of #, the solution
PO~Ol (/-t).
Now, let us consider the system (98) where the unknown are P ~blo and Polio and where we want to
find (L~°) - L o ) commuting with L~'. We use theorem 1 to define the projections /7 and ( I d - / 7 ) ,
respectively, on Image ~¢(I) and Ker ~¢.~x). In fact we want
/7(L(~° , - Lo) = 0 ,
(101)
then the system 98) reduces to
V-*lo(Id + Po*lo)-1( po.Wup° + po~q, po,,o + po£g p - Rio)
- - P - * ~ t P - 41o - P-"~Pv, Po - P - * ~ P o * 1 0 = 0,
(102)
/-/{(Id + Po#lo)-l(eo.LP~,Po + VoLP~,PoCxo+ Po.La~,P_Cblo)- L o ) = 0 .
We again solve (102) by the implicit function theorem, by choosing Po¢1o in a supplementary space of
Ker~¢ (1) in -W(Eo). It is not hard to check that the differential of the left-hand side of (102) with respect
to (P-~lO, Po¢1o) at the point 0, t~ --- 0, is for any (A, B) ~.W(Eo, E_) ×£a(Eo):
( A, B) ~ ( AL o - L_A, LoB - BLo).
(103)
The linear operator (103) is invertible since for the first component we already computed the inverse in
(19)1 with p = 1, and since for the second component we look for B in a supplementary space of K e r d (1).
Hence we can obtain P0#lO(/~) and P_~lo(/~) by identification of powers of/~ in (102), so L~°) follows
directly.
Having solved the problem for the linear terms, we can make the same analysis as in section 2.1, keeping
/~ at each step. We then obtain equations like (12), but with .W~,L~°), L~-) instead of L~a, L o, L , and #pq
depending on /~. We observe that (19) solves again the equations corresponding to (15), (16) since
exp(±L~°)t) increases slower than exp(L~-)t) decreases, when t ~ + oo. The only remaining problem is
with the homological equation
L~°)P0~pO[/~, X (p)] - DxPo~bpo[/x, X(e)] • L~°)X = Fp [/~, X (v)] + PoRp[tx, X(P)].
(104)
Introducing the projections H and I d - H, respectively, on Imagez~C(v) and on K e r d , ~p), and defining
L~°)= L o + L~1) ~Se(Eo)
where L~1)= 0(1#1),
eq. (104) reduces to
ad(V'PoCvo + / 7 ( [ Po~vo [/~, X(P)], L~I)x])= HPoR p.
(105)
120
C. Elphick et aL / Normal forms of singular vector fields
Choosing a supplementary space of K e r d ( P ) , this equation is uniquely solvable in Po~po since for It~l
small the linear operator acting on Po~po is a small perturbation of the now invertible operator ~ ( P ) . The
other part of (104) leads to an Fp regular in/~:
G=(II-Id){PoRp-[Po~po[t~,X~P'I,L~'X]} inKerag, (1'.
(106)
Hence theorem 6 is proved.
A.2 ~3~2 singularity
In this appendix we consider an example which is less elementary than the one presented in section 2.4.
Even though its codimension is 9, (hence it is very improbable physically) it has the interest to give a
counter-example to some a priori "reasonable" conjectures, for instance see remark A* in section 2.2.
Here we have
Lo=
'0
1
0
0
O'
0
0
1
0
0
0
0
0
0
0
0
0
0
0
1
,0
0
0
0
O,
(107)
If we note (xl, x 2, x3, X4, X5) = X and ( F 1, F2, F3, F 4, Fs) = F, and N* the differential operator defined by
~* = XI~
2 "4- X 2 ~ 3 "l- X 4 ~ 5 ,
(108)
then the partial differential system (41) becomes
~ * F 1 = 0,
N * F 2 = F1,
N * F 3 = F 2,
.@*F4 = 0,
~ * F 5 = F 4.
(109)
The characteristic system associated with (108) leads to the following 4 first integrals:
Z 1 = Xl,
Z 2 = x4,
Z 3 = x 2 -- 2xlx3,
Z 4 = X z X 4 - X l X 5.
(110)
Then F 1 has to be a function of Z 1, Z 2, Z 3, Z 4. We want F 1 to be a polynomial in (xl, x2, x3, x4, xs).
This does not imply that it is a polynomial in Z1, Z 2, Z 3, Z 4. A first nontrivial result is that F 1 is in fact a
polynomial in ( Z 1, Z 2, Z 3, Z 4, Z 5) where [6, 21],
Z s = x l x 2 + 2 x 3 x 2 - 2 x 2 x 4 x 5.
(111)
We remark that Z s = ( Z 2 - Z 2 Z 3 ) / Z I
is hence a first integral, polynomial of degree 3 in X but not
polynomial in ( Z 1, Z 2, Z 3, Z4).
Eq. (109)1 can then be solved by setting
F a = qOl(Zl, Z2, Z3, Z4, Zs) ,
where ~01 is a polynomial in its arguments.
(112)
121
C. Elphick et aL/ Normal forms of singular vector fields
We want now to solve (109)2. To wit let us write
F 1 = Xxl~l(X1, x4, Z3, Z4, Z5) -t- x41~2(x4, Z3, Z4, Z5) --{-Z41~3(Z3, Z4, Z5)
+ zs
4(z3, zs) +
(113)
where ( q~j; j = 1. . . . . 5} are polynomials in their arguments. The fact that F 1 is in the image of N* will lead
to 1/¢4 = 1~5 = 0.
Remark. Working with the scalar product defined in (25), it is clear that the adjoint N of N* is the
differential operator
~ = X 2 - ~ l q- X 3 - ~ 2 Of-X 5 OX 4,
whose kernel is formed by polynomials of the following type:
P ( x3, xs, x 2 - 2xlx3, x 2 x 5 - x 3 x , , x3x 2 + 2x1 x 2 - 2 x 2 x 4 x , ) .
So, the fact that F 1 is orthogonal to any such polynomial leads to
@4 = ~5 =
0.
The proof is not direct but also not very hard: let us write the solution F 2 of ~ * F 2 = F1 under the form
x2Z 5
X2
F2 = x2qq + xs~b2 + qlqJ3 + - - ~ 1 ~4 + ~-1 qJ5 + qOE(X1, x 4, Z 3, Z4),
(114)
where we only know that F 2 is a polynomial in (x 1. . . . . xs), ~kl. . . . . qJ5 are polynomials in their arguments
and ~2 is a rational function of its 4 arguments. We use the properties
~*x2=xl,
~*ql=Z4,
~*xs=x4,
~*(x225)=Z5,
x---7--
~ * ( X2 )
T11 = 1,
(115)
with ql = 2x3x4 - x2xs.
As a result of (114), we know that
a = x 2 Z s ~ 4 ( Z 3 , Zs) -]-x21~5(Z3) "Jr-Xl(]O2(x1, x4, Z3, Z4)
(116)
is a polynomial, vanishing at x 1 = 0.
This leads to
x2Z,
z,) + x2
,(xg) +
(o, x,, xg, x2x,, z,) = 0,
(117)
where we define ~ ( Z 1 , Z2, Z3, Z4, Zs) to be polynomial of its arguments equal to Xll~O2 (we now know
that it is a polynomial). Inspection of the monomials occurring in (117) immediately shows that
~k4 = q~5 = 0 and that Z 1 is in factor in q0~. We arrive at
F 1 = Xll~l -4- x41/J 2 --I- Z41~3,
F 2 = X21~l -Jr-X5~ 2 -t- qlq~3 + xlX x + x4X 2 + ZaX 3 + ZsX 4 + x s ( Z 3 ) ,
where (X j, J = 1 . . . . . 5 } are polynomials of the same arguments as (q~j }.
(118)
122
C. Elphick et al./ Normalforms of singular vectorfields
Let us now consider eq. (109)3 and set as for F 2,
x2Z 5
x # a 'r2
, + Z4'r3
q2t,t, 4 - x 2 x 1 4 - x 5 x 2 4 - q l X a 4 - - - - ~ l
F3 = x3@1 4- 214
12
X 4 4 - ~ l X S 4 - £ 0 3 ( X l , X4, Z a , Z 4 )
(119)
We wish to show that ~2 is divisible by x 4, +3 is divisible by Z 4, and X4 = X5 = 0, and that £03 is a
polynomial. We observe that
xg,,, [
q?
a = 2x4,~,,2\x4, Z3, Z 4, Z5) 4- -~4 ~3(Z 3, Z4, Z5) 4- _ _ x 4 ( Z 3 '
Z5)
12
4- .~1 x , ( Z 3 ) 4- £03(x1 ' x4 ' Z3,Z4)
(120)
is a polynomial in (Xl,..., 15). We deduce immediately that X114Z4£03(X1, X4, Z3, Z4) is a polynomial
£0~(Xl, x4, Z3, Z4, Z5). Multiplying (120) by x 4 and making x 4 = 0 shows that
~2(0, Z3, - xxx5, xtx~) = 0,
(121)
hence x 4 is in factor in q~2Multiplying (120) by Z 4 and making Z 4 = 0 shows that
qJ3(Z3,0, Zs) = 0
(since ql e 0, and Z3, Z5 are still independent),
hence Z 4 is in factor in q~3-Now we just make the same proof as above for ~k4 and q~5. Finally, we easily
obtain (changing notations)
el = x1+1 + xlq~2 + z,~,3,
F 2 = x2~ 1 4- x4151~2 "Jr qlZ4~3 4- x1X 1 "Jrx 4 x 2 4" Z 4 x 3 ,
F3 = x3~bl 4" " ~ 2
(122)
4- q-~*3 4- x2X1 4" x'X24"qlX34"£03'
where ~1, Xt are polynomials in (x 1, x 4, Z 3, Z 4, Z5), ~k2 and Xz are polynomials in (14, Z 3, Z 4, Zs), ~3
and X3 polynomials in ( Z 3, Z 4, Zs), £03 is a polynomial in (xt, x 4, Z 3, Z 4, Z5) and ql = 21314 - x=xs.
We can compute in the same way the two last components of F:
F 4 = 1101 4" 1402 4" Z403 ,
(123)
F 5 = x201+x502+ q103 + £05,
where { 0j, j = 1, 2, 3 } are polynomials of the same arguments as ~j and £05 is a polynomial in
(x1, x4, Z3, Z4, Zs).
Note that the normal form does not enter into the frame of remark A* of section 2.2 since
X2X - X1X5)2
(2x
~3
- x11,)
x, -
1(21314 -- X2X5) 2
X415
+ ,1,~ x~12
+X3
0
0
X2X 4 -- X1X 5
0
2X3X4 -- 12X 5
0
0
0
0
0
0
is not contained in (45).
+ 03
0
X2X 4 -- X l l 5
2x3x 4
-
-
X2X 5
C. EIphick et aL/ Normal forms of singular vectorfields
123
Finally let us give the explicit form for the rational fractions a j, j = 1,..., 5 characterizing the normal
form of ~3~.2. From (43) we obtain
F1
Xl0/1,
F 2 = x20/1 ÷
x10/2,
(124)
F 3 = x30/1 + x20/2 + x10/3,
F4 = x40/4,
F 5 = X50/4 "q- X40/5.
By equating (124) to (122) and (123) we obtain after some algebra the following expressions for the
aj, j = l . . . . . 5:
1
0/,= ~(x~4+2 + z~,,) + < ,
1 (X4X 2 ÷ Z4X3),
0/2 ~" -- X~ (x4Z41~2 ÷ x4Z4Z31113) ÷ ~1
0/,= 2x--(l[(Z ~,+ z,x~)q,2 + x~,Z~q,d - - Z , x2 +
(x,Z,x,
+%),
0l4 = 1 ( X 1 0 1 ÷ Z403) ÷ 02,
0/5 =
1
( Z 4 0 1 + ZsO,) + -~4 %.
Therefore in accordance with theorem 3 the rational fractions for f3fz are characterized by denominators x : , p < 3, and x~, q < 2, where the highest p(q) is the dimension of the critical subspace associated
~,ith the ~-3(~2) Jordan block.
4.3. (a) ~2 singularity. Proof of (49)
Since eq. (34) leads to
x 0Fx
~- 0,
aF2
X - ~ - "m-F1,
(125)
we obtain that Fl(X , y) = tp(x). Since F 1 is a polynomial in (x, y) then ~o is a polynomial in x. From
(125)2 we obtain
3F2
q0(x)
ay = x '
(126)
which is a polynomial. Hence ~ is divisible by x and can be written as
cp( x ) = xepx( x ),
(127)
124
C. Elphick et aL / Normal forms of singular vector fields
where ~0x is a polynomial in x. On solving (126) we obtain
F2(x , y) =ycp,(x) + ~P2(x)
and since F2 and ycpx(x ) are polynomials, also cp2 is a polynomial.
(b) ~3 singularity. Proof of (56)
Let us choose the new variables
ul=xl,
u2=x 2-2xlx3,
u3=x2
(128)
and define Fs(x 1, x2, x3) = / ~ ( u l , u2, u3) is an obvious way. Then the partial differential system (53) can be
written as
(129)
Eq. (129)1 gives
El(X1, x2, x3) = ~)(ul, u2).
Since F 1 is a polynomial in (xl, x2, x3) there exists n ~ Z ÷ such that
~nF 1
,~
~n~nep
= (-~xl) -g~2=O.
3x~
Hence it follows that rp is a polynomial in u 2. Therefore we can write
FI(X 1, X2, X3) ~- E F l k ( X 1 , X 2 ) X k ~- E I ~ ) k ( U l ) U k
k
k
from which it follows trivially that ~ok(ul) is a polynomial in u 1. This proves that qo is a polynomial in
(ul, u2) and can be written as ~0= ul~l(u1, u2) + q~l(u2) where qq(u2) = ~0(0, u2).
On solving (129)2 for if2 we obtain
x2
1
u
F2(xl, x2, x3) = x21~?x(ul, u2) --I--~-1i~1(u2) q- ~-11~( 1, u2),
(130)
where q~ is a polynomial in u 1, u 2 (the proof is the same as the one given for qo). Multiplying (130) by u 1
and making u 1 = 0 we obtain that q~l vanishes and that (l/u1) q, is a polynomial in (u 1, u2). By writing
q~/u 1 = ul~2( u 1, u2) + ~k2(u2) we obtain
F2(xl, x2, x3) = x2eol(Ul, u2) + ux~2(ul, u2) + ~2(u2),
(131)
where ~02(q~2) is a polynomial in (u 1, u2)((u2) ).
Finally, eq. (129)3 leads to
x2
1
F3(x 1, x2, x3) = x3cPl(Ul, u2) + x2qo2(Ul, u2) + ~-1~2(u2) + --UlX ( u l , u2) ,
(132)
C. Elphick et aL/ Normal forms of singular vectorfields
125
where X is a polynomial in (ul, u2) (see the proof of q0). Multiplying (132) by u I and making u I = 0 we
obtain ~k2 = 0. Hence (56) has been proved. For the ~2~2 singularity the proof is analogous to this one.
(c) ~2 singularity• Proof of (68)
Let us consider the new variables
•
U1 : 2IZ1,
z 2
U3 = l o ~ - - + l o g z l ,
21
U 2 = ZIZ 2 -- 2122,
which are independent first integrals of the characteristic system (64).
From (64) we obtain [20]
F1(
z2,
= z l r ( u l , u2,
(133)
since ziF1 is also a first integral. Let us prove that ¢p is a polynomial in (Ul, U2) and independent of u 3.
First we note that
O"~
=
(z:
z7
~"__y_~
~nep =
Z13u~
3
1
+ Zl
O)"
rl,
3)"
(
gl
~
(134)
'Zl
~
iw 3z 2 + i~ 3~2
)"
F1"
Since F 1 is a polynomial in (zl, z2, zx, ~2) it follows from (134) that q0 is a polynomial in (Ul, U2, U3).
Therefore qo is a sum of monomials u 1~u2~2u3~3which behaves as z2"~+~2(log Zl) ~3 for z 1 --->0¢ in R ÷. This is
not possible for a polynomial in z I except if a 3 = 0. Finally we can write
F I ( Z I ' Z2' ZI' Z2) = Z I ~ I ( U l , //2),
(135)
where q01 is a polynomial in (ul, U2).
Since F 2 -z2.~pl(ul, u2) and F 1 satisfy the same partial differential equation we immediately obtain
(67) and therefore (68).
Note added in proof*
It is worthwhile remarking that the normal form for ~3~2 instability (eqs. (122) and (123)) although is
complete it is not written in its minimal form, that means that the coefficient of a given term is not
uniquely determined (in other words the normal form contains some repeated terms). The non-minimality
of our normal form comes from the fact that Zx, Z2, Z3, Z4, Z 5 satisfy the relation Z 2 = Z1Z 5 + Z2Z3.
Therefore in order to obtain a minimal normal form we have to allow the functions F1, F2, F3, F4, F 5 to be
*We thank one of the referees for having called our attention about the non-minimality of the normal form given by (122) and
(123). We also thank the same referee for providing us the property (130) which was essential in the proof of minimallty.
126
C. Elphick et al./ Normal forms of singular vector fields
at most linear in Z 4. The minimal normal form is obtained as follows: we write
V4 = ~p~(Z1, Z2, Z3, Z , , Zs) + Z4q,2(Z1, Z2, Z3, Z4, Zs).
(136)
We easily find that q,~ is of the form x 1 % ( Z 1, Z2, Z3, Z s ) + x4qo2(Z1, Z2, Z3, Zs) since F 4 must be in
I m a g e ( ~ * ) . Hence it follows from (115) that
F 4 = XlqO1 "1- x4qo2 q- Z4q~2,
F, = x2q01 + xsfl02 + qx*2 + qo3(Z1, Z2, Z3, Z5).
(137)
Similarly we write
r 1 = ~po( Z , , Z 2, Z 3 , Zs) + Z4~b0(Z ,, Z 2, Z 3 , Z s ) .
(138)
We note that we can always write
z2, z3, z s ) + + 2 ( z 2 , z3, z s ) .
=
By imposing that F 1 ~ I m a g e ( ~ * ) and F 1 ~ I m a g e ( ( ~ * ) 2) we easily conclude that ~k2 is divisible by x 4.
Similar arguments show that ~0 has the form
q~o = xll~3(Z1, Z2, Z3, Z5) q-xIX,~d4(Z1, Z2, g3, Z5) q-x21p5 (Z2, Z3, Zs)-
(139)
Hence
F1 = xlz4~b 1 + x4z4qj 2 + xl~b3 + xlx4~k 4 + x4~ks.2
(140a)
Using (115) and imposing F 2 ~ I m a g e ( ~ * ) we obtain
F2 = x2Z4qJ1 + ( x 3 x ] - ~ x , x 5
) q~2+ Xzq~3+
+x~q'6(Z1, Z2, Z3, Zs) + x4q'v(Za, Z2, Z3, Z s ) ,
(140b)
1 2
r 3 = x 3 Z , + , + x 5 ( x 3 x 4 - ½XzXs) q~2 + x3q~ 3 + ½(XzXs) ~b, + ~xs~b 5
-FX2• 6 "b X5+ 7 -'F ~ 8 ( Z 1 , Z2, Z3, Z5).
(140c)
Although by construction the normal form defined by eqs. (137) and (140a, b, c) is minimal let us give
an explicit proof of minimality. First we note that the normal form has the form E 12
j= l f j v j - N , where
fj ~ K e r ( ~ * ) and vj, j = 1,12 are vectors satisfying eqs. (109). The normal form will be minimal if N = 0
implies fj = 0, Vj = 1,12. To prove this statement we observe that if P, Q are polynomials in Z 1, Z2, Z 3, Z s,
then*
P + Z4Q = 0
P = Q = 0.
(141)
Writing the normal form in vector form we see that we have to prove that
cPl
(xxt (14t
+ q~2
X2
+ ~2
X5
+ %
ql
(0) (:t
=
(142)
1
implies cp1 -- ~P2 = % = q'2 = 0. Using (141) the first component of (142) gives qh = 0, cp1 = x4% cp2 = - xdp
for some cp. Therefore the second component" of (142) reads % + Z4ep = 0 which by (141) leads to % = 0,
cp = 0. We also have to prove that F 1 = F 2 = F 3 = 0 implies ~j = 0, j = 1 . . . . . 8. Using (141) we obtain from
C. Elphick et al./ Normal forms of singular vector fields
127
F~ = 0 that
(143a)
xl@ , + x4@ 2 = 0,
XlI~3 q- XlX41~4 "~-X2~5
O.
:
(143b)
Since ~k2 does not depend on x 1 (143a) gives q~a = ~k2 = 0. Similarly since lk5 does not depend on x 1 (143b)
gives q~5 = 0 and
(144)
~3 = --X41~4"
U s i n g (144) the condition F 2 = 0 gives
XII~6 "+" X4~J 7 -- Z 4 ~ 4
:
0
which b y (141) leads to
~4 : 0,
~6 : --X4~'
~7 : Xa~b
(145)
for some @. U s i n g (145) we obtain that F 3 = 0 reads
~b8 - Z4~ = 0
(146)
a n d therefore b y (141) ~b8 = @ = 0 which finally proves the minimality of the n o r m a l f o r m given b y eqs.
(137) and (140a, b, c).
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